7.1 Motivations

7.1.1 How do the twistors emerge?

In the Newtonian theory of gravity the mass contained in some finite 3-volume can be
expressed as the flux integral of the gravitational field strength on the boundary :

where is the gravitational potential and is the outward directed unit normal to . If is
deformed in through a source-free region, then the mass does not change. Thus the mass is
analogous to charge in electrostatics.

In the weak field (linear) approximation of general relativity on Minkowski spacetime the source of the
gravitational field, i.e. the linearized energy-momentum tensor, is still analogous to charge. In fact, the
total energy-momentum and angular momentum of the source can be expressed as appropriate 2-surface
integrals of the curvature at infinity [349]. Thus it is natural to expect that the energy-momentum and
angular momentum of the source in a finite 3-volume , given by Equation (5), can also be expressed as
the charge integral of the curvature on the 2-surface . However, the curvature tensor can
be integrated on only if at least one pair of its indices is annihilated by some tensor via
contraction, i.e. according to Equation (15) if some is chosen and . To
simplify the subsequent analysis will be chosen to be anti-self-dual: with
11.
Thus our claim is to find an appropriate spinor field on such that

Since the dual of the exterior derivative of the integrand on the right, and, by Einstein’s equations, the dual
of the times the integrand on the left, respectively, is

This equation in its symmetrized form, , is the valence 2 twistor equation, a specific
example for the general twistor equation for . Thus, as could be
expected, depends on the Killing vector , and, in fact, can be recovered from as
. Thus plays the role of a potential for the Killing vector . However, as a
consequence of Equation (46), is a self-dual Killing 1-form in the sense that its derivative is a
self-dual (or s.d.) 2-form: In fact, the general solution of equation (46) and the corresponding Killing vector
are

where , , and are constant spinors, using the notation , where
is a constant spin frame (the ‘Cartesian spin frame’) and are the standard Pauli
matrices (divided by ). These yield that is, in fact, self-dual, , and is
a translation and generates self-dual rotations. Then , implying
that the charges corresponding to are vanishing; the four components of the quasi-local
energy-momentum correspond to the reals, and the spatial angular momentum and centre-of-mass
are combined into the three complex components of the self-dual angular momentum , generated by
.

7.1.2 Twistor space and the kinematical twistor

Recall that the space of the contravariant valence 1 twistors of Minkowski spacetime is the set of the
pairs of spinor fields, which solve the so-called valence 1 twistor equation
. If is a solution of this equation, then is a solution
of the corresponding equation in the conformally rescaled spacetime, where and
is the conformal factor. In general the twistor equation has only the trivial solution, but in
the (conformal) Minkowski spacetime it has a four complex parameter family of solutions. Its
general solution in the Minkowski spacetime is , where and are
constant spinors. Thus the space of valence 1 twistors, the so-called twistor-space, is
4-complex-dimensional, and hence has the structure . admits a natural Hermitianscalar product: If is another twistor, then . Its
signature is , it is conformally invariant, , and it is
constant on Minkowski spacetime. The metric defines a natural isomorphism between the
complex conjugate twistor space, , and the dual twistor space, , by
. This makes it possible to use only twistors with unprimed indices. In particular, the
complex conjugate of the covariant valence 2 twistor can be represented by the
so-called conjugate twistor . We should mention two special, higher valence
twistors. The first is the so-called infinity twistor. This and its conjugate are given explicitly by

The other is the completely anti-symmetric twistor , whose component in an
-orthonormal basis is required to be one. The only non-vanishing spinor parts of are those
with two primed and two unprimed spinor indices: , ,
, …. Thus for any four twistors , , the determinant of the
matrix whose -th column is , where the , …, are the components of
the spinors and in some spin frame, is

where is the totally antisymmetric Levi-Civita symbol. Then and are dual to each other
in the sense that , and by the simplicity of one has .

The solution of the valence 2 twistor equation, given by Equation (47), can always be written as
a linear combination of the symmetrized product of the solutions and of the valence 1
twistor equation. defines uniquely a symmetric twistor (see for example [313]). Its spinor
parts are

However, Equation (43) can be interpreted as a -linear mapping of into , i.e. Equation (43)
defines a dual twistor, the (symmetric) kinematical twistor, which therefore has the structure

Thus the quasi-local energy-momentum and self-dual angular momentum of the source are certain spinorparts of the kinematical twistor. In contrast to the ten complex components of a general symmetric twistor,
it has only ten real components as a consequence of its structure (its spinor part is identically zero)
and the reality of . These properties can be reformulated by the infinity twistor and the Hermitian
metric as conditions on : The vanishing of the spinor part is equivalent to
and the energy momentum is the part of the kinematical twistor, while the whole
reality condition (ensuring both and the reality of the energy-momentum) is equivalent to

Using the conjugate twistors this can be rewritten (and, in fact, usually it is written) as
. Finally, the quasi-local mass can also be
expressed by the kinematical twistor as its Hermitian norm [307] or as its determinant [371]:

Thus, to summarize, the various spinor parts of the kinematical twistor are the energy-momentum
and s.d. angular momentum. However, additional structures, namely the infinity twistor and the Hermitian
scalar product, were needed to be able to ‘isolate’ its energy-momentum and angular momentum parts, and,
in particular, to define the mass. Furthermore, the Hermiticity condition ensuring to have the
correct number of components (ten reals) were also formulated in terms of these additional
structures.

"Quasi-Local Energy-Momentum and Angular Momentum in
GR: A Review Article"